Neural Optimal Transport

Neural Optimal Transport with General Cost Functionals

“…Weak formulation (3) can be viewed as a generalization of strong formulation (2) when C(x, µ) = Y c(x, y)dµ(y). An example of a weak cost function is the γ-weak (γ > 0) quadratic cost [42,Eq. 5], [1, 5.2], [27, 5.2]:…”

“…Following the notation of [42], we say that functions T are stochastic maps (Figure 3). We say that T * is a stochastic OT map if it represents some OT plan π * , i.e., T * (x, •) S = π * (•|x) holds P-almost surely for all x ∈ X .…”

“…Optimal transport (OT) is a powerful framework to solve mass-moving and generative modeling problems for data distributions. Recent works [42,64,40,19,17] propose scalable neural methods to compute OT plans (or maps). They show that the learned transport plan (or map) can be used directly as the generative model in data synthesis [64] and unpaired learning [42,64,17,23].…”

“…Recent works [42,64,40,19,17] propose scalable neural methods to compute OT plans (or maps). They show that the learned transport plan (or map) can be used directly as the generative model in data synthesis [64] and unpaired learning [42,64,17,23]. Compared to WGANs [8] which employ OT cost as the loss for generator [64,3], these methods provide better flexibility: the properties of the learned model can be controlled by the transport cost function.…”

“…Existing neural OT plan (or map) methods consider distance-based cost functions, e.g., weak or strong quadratic costs [42,19,23]. These costs are suitable for unpaired restoration and style translation.…”